// === Sylvester ===
// Vector and Matrix mathematics modules for JavaScript
// Copyright (c) 2007 James Coglan
// 
// Permission is hereby granted, free of charge, to any person obtaining
// a copy of this software and associated documentation files (the "Software"),
// to deal in the Software without restriction, including without limitation
// the rights to use, copy, modify, merge, publish, distribute, sublicense,
// and/or sell copies of the Software, and to permit persons to whom the
// Software is furnished to do so, subject to the following conditions:
// 
// The above copyright notice and this permission notice shall be included
// in all copies or substantial portions of the Software.
// 
// THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS
// OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
// FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL
// THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
// LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING
// FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER
// DEALINGS IN THE SOFTWARE.

var Sylvester = {
    version: '0.1.3',
    precision: 1e-6
  };
  
  function Vector() {}
  Vector.prototype = {
  
    // Returns element i of the vector
    e: function(i) {
      return (i < 1 || i > this.elements.length) ? null : this.elements[i-1];
    },
  
    // Returns the number of elements the vector has
    dimensions: function() {
      return this.elements.length;
    },
  
    // Returns the modulus ('length') of the vector
    modulus: function() {
      return Math.sqrt(this.dot(this));
    },
  
    // Returns true iff the vector is equal to the argument
    eql: function(vector) {
      var n = this.elements.length;
      var V = vector.elements || vector;
      if (n != V.length) { return false; }
      do {
        if (Math.abs(this.elements[n-1] - V[n-1]) > Sylvester.precision) { return false; }
      } while (--n);
      return true;
    },
  
    // Returns a copy of the vector
    dup: function() {
      return Vector.create(this.elements);
    },
  
    // Maps the vector to another vector according to the given function
    map: function(fn) {
      var elements = [];
      this.each(function(x, i) {
        elements.push(fn(x, i));
      });
      return Vector.create(elements);
    },
    
    // Calls the iterator for each element of the vector in turn
    each: function(fn) {
      var n = this.elements.length, k = n, i;
      do { i = k - n;
        fn(this.elements[i], i+1);
      } while (--n);
    },
  
    // Returns a new vector created by normalizing the receiver
    toUnitVector: function() {
      var r = this.modulus();
      if (r === 0) { return this.dup(); }
      return this.map(function(x) { return x/r; });
    },
  
    // Returns the angle between the vector and the argument (also a vector)
    angleFrom: function(vector) {
      var V = vector.elements || vector;
      var n = this.elements.length, k = n, i;
      if (n != V.length) { return null; }
      var dot = 0, mod1 = 0, mod2 = 0;
      // Work things out in parallel to save time
      this.each(function(x, i) {
        dot += x * V[i-1];
        mod1 += x * x;
        mod2 += V[i-1] * V[i-1];
      });
      mod1 = Math.sqrt(mod1); mod2 = Math.sqrt(mod2);
      if (mod1*mod2 === 0) { return null; }
      var theta = dot / (mod1*mod2);
      if (theta < -1) { theta = -1; }
      if (theta > 1) { theta = 1; }
      return Math.acos(theta);
    },
  
    // Returns true iff the vector is parallel to the argument
    isParallelTo: function(vector) {
      var angle = this.angleFrom(vector);
      return (angle === null) ? null : (angle <= Sylvester.precision);
    },
  
    // Returns true iff the vector is antiparallel to the argument
    isAntiparallelTo: function(vector) {
      var angle = this.angleFrom(vector);
      return (angle === null) ? null : (Math.abs(angle - Math.PI) <= Sylvester.precision);
    },
  
    // Returns true iff the vector is perpendicular to the argument
    isPerpendicularTo: function(vector) {
      var dot = this.dot(vector);
      return (dot === null) ? null : (Math.abs(dot) <= Sylvester.precision);
    },
  
    // Returns the result of adding the argument to the vector
    add: function(vector) {
      var V = vector.elements || vector;
      if (this.elements.length != V.length) { return null; }
      return this.map(function(x, i) { return x + V[i-1]; });
    },
  
    // Returns the result of subtracting the argument from the vector
    subtract: function(vector) {
      var V = vector.elements || vector;
      if (this.elements.length != V.length) { return null; }
      return this.map(function(x, i) { return x - V[i-1]; });
    },
  
    // Returns the result of multiplying the elements of the vector by the argument
    multiply: function(k) {
      return this.map(function(x) { return x*k; });
    },
  
    x: function(k) { return this.multiply(k); },
  
    // Returns the scalar product of the vector with the argument
    // Both vectors must have equal dimensionality
    dot: function(vector) {
      var V = vector.elements || vector;
      var i, product = 0, n = this.elements.length;
      if (n != V.length) { return null; }
      do { product += this.elements[n-1] * V[n-1]; } while (--n);
      return product;
    },
  
    // Returns the vector product of the vector with the argument
    // Both vectors must have dimensionality 3
    cross: function(vector) {
      var B = vector.elements || vector;
      if (this.elements.length != 3 || B.length != 3) { return null; }
      var A = this.elements;
      return Vector.create([
        (A[1] * B[2]) - (A[2] * B[1]),
        (A[2] * B[0]) - (A[0] * B[2]),
        (A[0] * B[1]) - (A[1] * B[0])
      ]);
    },
  
    // Returns the (absolute) largest element of the vector
    max: function() {
      var m = 0, n = this.elements.length, k = n, i;
      do { i = k - n;
        if (Math.abs(this.elements[i]) > Math.abs(m)) { m = this.elements[i]; }
      } while (--n);
      return m;
    },
  
    // Returns the index of the first match found
    indexOf: function(x) {
      var index = null, n = this.elements.length, k = n, i;
      do { i = k - n;
        if (index === null && this.elements[i] == x) {
          index = i + 1;
        }
      } while (--n);
      return index;
    },
  
    // Returns a diagonal matrix with the vector's elements as its diagonal elements
    toDiagonalMatrix: function() {
      return Matrix.Diagonal(this.elements);
    },
  
    // Returns the result of rounding the elements of the vector
    round: function() {
      return this.map(function(x) { return Math.round(x); });
    },
  
    // Returns a copy of the vector with elements set to the given value if they
    // differ from it by less than Sylvester.precision
    snapTo: function(x) {
      return this.map(function(y) {
        return (Math.abs(y - x) <= Sylvester.precision) ? x : y;
      });
    },
  
    // Returns the vector's distance from the argument, when considered as a point in space
    distanceFrom: function(obj) {
      if (obj.anchor) { return obj.distanceFrom(this); }
      var V = obj.elements || obj;
      if (V.length != this.elements.length) { return null; }
      var sum = 0, part;
      this.each(function(x, i) {
        part = x - V[i-1];
        sum += part * part;
      });
      return Math.sqrt(sum);
    },
  
    // Returns true if the vector is point on the given line
    liesOn: function(line) {
      return line.contains(this);
    },
  
    // Return true iff the vector is a point in the given plane
    liesIn: function(plane) {
      return plane.contains(this);
    },
  
    // Rotates the vector about the given object. The object should be a 
    // point if the vector is 2D, and a line if it is 3D. Be careful with line directions!
    rotate: function(t, obj) {
      var V, R, x, y, z;
      switch (this.elements.length) {
        case 2:
          V = obj.elements || obj;
          if (V.length != 2) { return null; }
          R = Matrix.Rotation(t).elements;
          x = this.elements[0] - V[0];
          y = this.elements[1] - V[1];
          return Vector.create([
            V[0] + R[0][0] * x + R[0][1] * y,
            V[1] + R[1][0] * x + R[1][1] * y
          ]);
          break;
        case 3:
          if (!obj.direction) { return null; }
          var C = obj.pointClosestTo(this).elements;
          R = Matrix.Rotation(t, obj.direction).elements;
          x = this.elements[0] - C[0];
          y = this.elements[1] - C[1];
          z = this.elements[2] - C[2];
          return Vector.create([
            C[0] + R[0][0] * x + R[0][1] * y + R[0][2] * z,
            C[1] + R[1][0] * x + R[1][1] * y + R[1][2] * z,
            C[2] + R[2][0] * x + R[2][1] * y + R[2][2] * z
          ]);
          break;
        default:
          return null;
      }
    },
  
    // Returns the result of reflecting the point in the given point, line or plane
    reflectionIn: function(obj) {
      if (obj.anchor) {
        // obj is a plane or line
        var P = this.elements.slice();
        var C = obj.pointClosestTo(P).elements;
        return Vector.create([C[0] + (C[0] - P[0]), C[1] + (C[1] - P[1]), C[2] + (C[2] - (P[2] || 0))]);
      } else {
        // obj is a point
        var Q = obj.elements || obj;
        if (this.elements.length != Q.length) { return null; }
        return this.map(function(x, i) { return Q[i-1] + (Q[i-1] - x); });
      }
    },
  
    // Utility to make sure vectors are 3D. If they are 2D, a zero z-component is added
    to3D: function() {
      var V = this.dup();
      switch (V.elements.length) {
        case 3: break;
        case 2: V.elements.push(0); break;
        default: return null;
      }
      return V;
    },
  
    // Returns a string representation of the vector
    inspect: function() {
      return '[' + this.elements.join(', ') + ']';
    },
  
    // Set vector's elements from an array
    setElements: function(els) {
      this.elements = (els.elements || els).slice();
      return this;
    }
  };
    
  // Constructor function
  Vector.create = function(elements) {
    var V = new Vector();
    return V.setElements(elements);
  };
  
  // i, j, k unit vectors
  Vector.i = Vector.create([1,0,0]);
  Vector.j = Vector.create([0,1,0]);
  Vector.k = Vector.create([0,0,1]);
  
  // Random vector of size n
  Vector.Random = function(n) {
    var elements = [];
    do { elements.push(Math.random());
    } while (--n);
    return Vector.create(elements);
  };
  
  // Vector filled with zeros
  Vector.Zero = function(n) {
    var elements = [];
    do { elements.push(0);
    } while (--n);
    return Vector.create(elements);
  };
  
  
  
  function Matrix() {}
  Matrix.prototype = {
  
    // Returns element (i,j) of the matrix
    e: function(i,j) {
      if (i < 1 || i > this.elements.length || j < 1 || j > this.elements[0].length) { return null; }
      return this.elements[i-1][j-1];
    },
  
    // Returns row k of the matrix as a vector
    row: function(i) {
      if (i > this.elements.length) { return null; }
      return Vector.create(this.elements[i-1]);
    },
  
    // Returns column k of the matrix as a vector
    col: function(j) {
      if (j > this.elements[0].length) { return null; }
      var col = [], n = this.elements.length, k = n, i;
      do { i = k - n;
        col.push(this.elements[i][j-1]);
      } while (--n);
      return Vector.create(col);
    },
  
    // Returns the number of rows/columns the matrix has
    dimensions: function() {
      return {rows: this.elements.length, cols: this.elements[0].length};
    },
  
    // Returns the number of rows in the matrix
    rows: function() {
      return this.elements.length;
    },
  
    // Returns the number of columns in the matrix
    cols: function() {
      return this.elements[0].length;
    },
  
    // Returns true iff the matrix is equal to the argument. You can supply
    // a vector as the argument, in which case the receiver must be a
    // one-column matrix equal to the vector.
    eql: function(matrix) {
      var M = matrix.elements || matrix;
      if (typeof(M[0][0]) == 'undefined') { M = Matrix.create(M).elements; }
      if (this.elements.length != M.length ||
          this.elements[0].length != M[0].length) { return false; }
      var ni = this.elements.length, ki = ni, i, nj, kj = this.elements[0].length, j;
      do { i = ki - ni;
        nj = kj;
        do { j = kj - nj;
          if (Math.abs(this.elements[i][j] - M[i][j]) > Sylvester.precision) { return false; }
        } while (--nj);
      } while (--ni);
      return true;
    },
  
    // Returns a copy of the matrix
    dup: function() {
      return Matrix.create(this.elements);
    },
  
    // Maps the matrix to another matrix (of the same dimensions) according to the given function
    map: function(fn) {
      var els = [], ni = this.elements.length, ki = ni, i, nj, kj = this.elements[0].length, j;
      do { i = ki - ni;
        nj = kj;
        els[i] = [];
        do { j = kj - nj;
          els[i][j] = fn(this.elements[i][j], i + 1, j + 1);
        } while (--nj);
      } while (--ni);
      return Matrix.create(els);
    },
  
    // Returns true iff the argument has the same dimensions as the matrix
    isSameSizeAs: function(matrix) {
      var M = matrix.elements || matrix;
      if (typeof(M[0][0]) == 'undefined') { M = Matrix.create(M).elements; }
      return (this.elements.length == M.length &&
          this.elements[0].length == M[0].length);
    },
  
    // Returns the result of adding the argument to the matrix
    add: function(matrix) {
      var M = matrix.elements || matrix;
      if (typeof(M[0][0]) == 'undefined') { M = Matrix.create(M).elements; }
      if (!this.isSameSizeAs(M)) { return null; }
      return this.map(function(x, i, j) { return x + M[i-1][j-1]; });
    },
  
    // Returns the result of subtracting the argument from the matrix
    subtract: function(matrix) {
      var M = matrix.elements || matrix;
      if (typeof(M[0][0]) == 'undefined') { M = Matrix.create(M).elements; }
      if (!this.isSameSizeAs(M)) { return null; }
      return this.map(function(x, i, j) { return x - M[i-1][j-1]; });
    },
  
    // Returns true iff the matrix can multiply the argument from the left
    canMultiplyFromLeft: function(matrix) {
      var M = matrix.elements || matrix;
      if (typeof(M[0][0]) == 'undefined') { M = Matrix.create(M).elements; }
      // this.columns should equal matrix.rows
      return (this.elements[0].length == M.length);
    },
  
    // Returns the result of multiplying the matrix from the right by the argument.
    // If the argument is a scalar then just multiply all the elements. If the argument is
    // a vector, a vector is returned, which saves you having to remember calling
    // col(1) on the result.
    multiply: function(matrix) {
      if (!matrix.elements) {
        return this.map(function(x) { return x * matrix; });
      }
      var returnVector = matrix.modulus ? true : false;
      var M = matrix.elements || matrix;
      if (typeof(M[0][0]) == 'undefined') { M = Matrix.create(M).elements; }
      if (!this.canMultiplyFromLeft(M)) { return null; }
      var ni = this.elements.length, ki = ni, i, nj, kj = M[0].length, j;
      var cols = this.elements[0].length, elements = [], sum, nc, c;
      do { i = ki - ni;
        elements[i] = [];
        nj = kj;
        do { j = kj - nj;
          sum = 0;
          nc = cols;
          do { c = cols - nc;
            sum += this.elements[i][c] * M[c][j];
          } while (--nc);
          elements[i][j] = sum;
        } while (--nj);
      } while (--ni);
      var M = Matrix.create(elements);
      return returnVector ? M.col(1) : M;
    },
  
    x: function(matrix) { return this.multiply(matrix); },
  
    // Returns a submatrix taken from the matrix
    // Argument order is: start row, start col, nrows, ncols
    // Element selection wraps if the required index is outside the matrix's bounds, so you could
    // use this to perform row/column cycling or copy-augmenting.
    minor: function(a, b, c, d) {
      var elements = [], ni = c, i, nj, j;
      var rows = this.elements.length, cols = this.elements[0].length;
      do { i = c - ni;
        elements[i] = [];
        nj = d;
        do { j = d - nj;
          elements[i][j] = this.elements[(a+i-1)%rows][(b+j-1)%cols];
        } while (--nj);
      } while (--ni);
      return Matrix.create(elements);
    },
  
    // Returns the transpose of the matrix
    transpose: function() {
      var rows = this.elements.length, cols = this.elements[0].length;
      var elements = [], ni = cols, i, nj, j;
      do { i = cols - ni;
        elements[i] = [];
        nj = rows;
        do { j = rows - nj;
          elements[i][j] = this.elements[j][i];
        } while (--nj);
      } while (--ni);
      return Matrix.create(elements);
    },
  
    // Returns true iff the matrix is square
    isSquare: function() {
      return (this.elements.length == this.elements[0].length);
    },
  
    // Returns the (absolute) largest element of the matrix
    max: function() {
      var m = 0, ni = this.elements.length, ki = ni, i, nj, kj = this.elements[0].length, j;
      do { i = ki - ni;
        nj = kj;
        do { j = kj - nj;
          if (Math.abs(this.elements[i][j]) > Math.abs(m)) { m = this.elements[i][j]; }
        } while (--nj);
      } while (--ni);
      return m;
    },
  
    // Returns the indeces of the first match found by reading row-by-row from left to right
    indexOf: function(x) {
      var index = null, ni = this.elements.length, ki = ni, i, nj, kj = this.elements[0].length, j;
      do { i = ki - ni;
        nj = kj;
        do { j = kj - nj;
          if (this.elements[i][j] == x) { return {i: i+1, j: j+1}; }
        } while (--nj);
      } while (--ni);
      return null;
    },
  
    // If the matrix is square, returns the diagonal elements as a vector.
    // Otherwise, returns null.
    diagonal: function() {
      if (!this.isSquare) { return null; }
      var els = [], n = this.elements.length, k = n, i;
      do { i = k - n;
        els.push(this.elements[i][i]);
      } while (--n);
      return Vector.create(els);
    },
  
    // Make the matrix upper (right) triangular by Gaussian elimination.
    // This method only adds multiples of rows to other rows. No rows are
    // scaled up or switched, and the determinant is preserved.
    toRightTriangular: function() {
      var M = this.dup(), els;
      var n = this.elements.length, k = n, i, np, kp = this.elements[0].length, p;
      do { i = k - n;
        if (M.elements[i][i] == 0) {
          for (j = i + 1; j < k; j++) {
            if (M.elements[j][i] != 0) {
              els = []; np = kp;
              do { p = kp - np;
                els.push(M.elements[i][p] + M.elements[j][p]);
              } while (--np);
              M.elements[i] = els;
              break;
            }
          }
        }
        if (M.elements[i][i] != 0) {
          for (j = i + 1; j < k; j++) {
            var multiplier = M.elements[j][i] / M.elements[i][i];
            els = []; np = kp;
            do { p = kp - np;
              // Elements with column numbers up to an including the number
              // of the row that we're subtracting can safely be set straight to
              // zero, since that's the point of this routine and it avoids having
              // to loop over and correct rounding errors later
              els.push(p <= i ? 0 : M.elements[j][p] - M.elements[i][p] * multiplier);
            } while (--np);
            M.elements[j] = els;
          }
        }
      } while (--n);
      return M;
    },
  
    toUpperTriangular: function() { return this.toRightTriangular(); },
  
    // Returns the determinant for square matrices
    determinant: function() {
      if (!this.isSquare()) { return null; }
      var M = this.toRightTriangular();
      var det = M.elements[0][0], n = M.elements.length - 1, k = n, i;
      do { i = k - n + 1;
        det = det * M.elements[i][i];
      } while (--n);
      return det;
    },
  
    det: function() { return this.determinant(); },
  
    // Returns true iff the matrix is singular
    isSingular: function() {
      return (this.isSquare() && this.determinant() === 0);
    },
  
    // Returns the trace for square matrices
    trace: function() {
      if (!this.isSquare()) { return null; }
      var tr = this.elements[0][0], n = this.elements.length - 1, k = n, i;
      do { i = k - n + 1;
        tr += this.elements[i][i];
      } while (--n);
      return tr;
    },
  
    tr: function() { return this.trace(); },
  
    // Returns the rank of the matrix
    rank: function() {
      var M = this.toRightTriangular(), rank = 0;
      var ni = this.elements.length, ki = ni, i, nj, kj = this.elements[0].length, j;
      do { i = ki - ni;
        nj = kj;
        do { j = kj - nj;
          if (Math.abs(M.elements[i][j]) > Sylvester.precision) { rank++; break; }
        } while (--nj);
      } while (--ni);
      return rank;
    },
    
    rk: function() { return this.rank(); },
  
    // Returns the result of attaching the given argument to the right-hand side of the matrix
    augment: function(matrix) {
      var M = matrix.elements || matrix;
      if (typeof(M[0][0]) == 'undefined') { M = Matrix.create(M).elements; }
      var T = this.dup(), cols = T.elements[0].length;
      var ni = T.elements.length, ki = ni, i, nj, kj = M[0].length, j;
      if (ni != M.length) { return null; }
      do { i = ki - ni;
        nj = kj;
        do { j = kj - nj;
          T.elements[i][cols + j] = M[i][j];
        } while (--nj);
      } while (--ni);
      return T;
    },
  
    // Returns the inverse (if one exists) using Gauss-Jordan
    inverse: function() {
      if (!this.isSquare() || this.isSingular()) { return null; }
      var ni = this.elements.length, ki = ni, i, j;
      var M = this.augment(Matrix.I(ni)).toRightTriangular();
      var np, kp = M.elements[0].length, p, els, divisor;
      var inverse_elements = [], new_element;
      // Matrix is non-singular so there will be no zeros on the diagonal
      // Cycle through rows from last to first
      do { i = ni - 1;
        // First, normalise diagonal elements to 1
        els = []; np = kp;
        inverse_elements[i] = [];
        divisor = M.elements[i][i];
        do { p = kp - np;
          new_element = M.elements[i][p] / divisor;
          els.push(new_element);
          // Shuffle of the current row of the right hand side into the results
          // array as it will not be modified by later runs through this loop
          if (p >= ki) { inverse_elements[i].push(new_element); }
        } while (--np);
        M.elements[i] = els;
        // Then, subtract this row from those above it to
        // give the identity matrix on the left hand side
        for (j = 0; j < i; j++) {
          els = []; np = kp;
          do { p = kp - np;
            els.push(M.elements[j][p] - M.elements[i][p] * M.elements[j][i]);
          } while (--np);
          M.elements[j] = els;
        }
      } while (--ni);
      return Matrix.create(inverse_elements);
    },
  
    inv: function() { return this.inverse(); },
  
    // Returns the result of rounding all the elements
    round: function() {
      return this.map(function(x) { return Math.round(x); });
    },
  
    // Returns a copy of the matrix with elements set to the given value if they
    // differ from it by less than Sylvester.precision
    snapTo: function(x) {
      return this.map(function(p) {
        return (Math.abs(p - x) <= Sylvester.precision) ? x : p;
      });
    },
  
    // Returns a string representation of the matrix
    inspect: function() {
      var matrix_rows = [];
      var n = this.elements.length, k = n, i;
      do { i = k - n;
        matrix_rows.push(Vector.create(this.elements[i]).inspect());
      } while (--n);
      return matrix_rows.join('\n');
    },
  
    // Set the matrix's elements from an array. If the argument passed
    // is a vector, the resulting matrix will be a single column.
    setElements: function(els) {
      var i, elements = els.elements || els;
      if (typeof(elements[0][0]) != 'undefined') {
        var ni = elements.length, ki = ni, nj, kj, j;
        this.elements = [];
        do { i = ki - ni;
          nj = elements[i].length; kj = nj;
          this.elements[i] = [];
          do { j = kj - nj;
            this.elements[i][j] = elements[i][j];
          } while (--nj);
        } while(--ni);
        return this;
      }
      var n = elements.length, k = n;
      this.elements = [];
      do { i = k - n;
        this.elements.push([elements[i]]);
      } while (--n);
      return this;
    }
  };
  
  // Constructor function
  Matrix.create = function(elements) {
    var M = new Matrix();
    return M.setElements(elements);
  };
  
  // Identity matrix of size n
  Matrix.I = function(n) {
    var els = [], k = n, i, nj, j;
    do { i = k - n;
      els[i] = []; nj = k;
      do { j = k - nj;
        els[i][j] = (i == j) ? 1 : 0;
      } while (--nj);
    } while (--n);
    return Matrix.create(els);
  };
  
  // Diagonal matrix - all off-diagonal elements are zero
  Matrix.Diagonal = function(elements) {
    var n = elements.length, k = n, i;
    var M = Matrix.I(n);
    do { i = k - n;
      M.elements[i][i] = elements[i];
    } while (--n);
    return M;
  };
  
  // Rotation matrix about some axis. If no axis is
  // supplied, assume we're after a 2D transform
  Matrix.Rotation = function(theta, a) {
    if (!a) {
      return Matrix.create([
        [Math.cos(theta),  -Math.sin(theta)],
        [Math.sin(theta),   Math.cos(theta)]
      ]);
    }
    var axis = a.dup();
    if (axis.elements.length != 3) { return null; }
    var mod = axis.modulus();
    var x = axis.elements[0]/mod, y = axis.elements[1]/mod, z = axis.elements[2]/mod;
    var s = Math.sin(theta), c = Math.cos(theta), t = 1 - c;
    // Formula derived here: http://www.gamedev.net/reference/articles/article1199.asp
    // That proof rotates the co-ordinate system so theta
    // becomes -theta and sin becomes -sin here.
    return Matrix.create([
      [ t*x*x + c, t*x*y - s*z, t*x*z + s*y ],
      [ t*x*y + s*z, t*y*y + c, t*y*z - s*x ],
      [ t*x*z - s*y, t*y*z + s*x, t*z*z + c ]
    ]);
  };
  
  // Special case rotations
  Matrix.RotationX = function(t) {
    var c = Math.cos(t), s = Math.sin(t);
    return Matrix.create([
      [  1,  0,  0 ],
      [  0,  c, -s ],
      [  0,  s,  c ]
    ]);
  };
  Matrix.RotationY = function(t) {
    var c = Math.cos(t), s = Math.sin(t);
    return Matrix.create([
      [  c,  0,  s ],
      [  0,  1,  0 ],
      [ -s,  0,  c ]
    ]);
  };
  Matrix.RotationZ = function(t) {
    var c = Math.cos(t), s = Math.sin(t);
    return Matrix.create([
      [  c, -s,  0 ],
      [  s,  c,  0 ],
      [  0,  0,  1 ]
    ]);
  };
  
  // Random matrix of n rows, m columns
  Matrix.Random = function(n, m) {
    return Matrix.Zero(n, m).map(
      function() { return Math.random(); }
    );
  };
  
  // Matrix filled with zeros
  Matrix.Zero = function(n, m) {
    var els = [], ni = n, i, nj, j;
    do { i = n - ni;
      els[i] = [];
      nj = m;
      do { j = m - nj;
        els[i][j] = 0;
      } while (--nj);
    } while (--ni);
    return Matrix.create(els);
  };
  
  
  
  function Line() {}
  Line.prototype = {
  
    // Returns true if the argument occupies the same space as the line
    eql: function(line) {
      return (this.isParallelTo(line) && this.contains(line.anchor));
    },
  
    // Returns a copy of the line
    dup: function() {
      return Line.create(this.anchor, this.direction);
    },
  
    // Returns the result of translating the line by the given vector/array
    translate: function(vector) {
      var V = vector.elements || vector;
      return Line.create([
        this.anchor.elements[0] + V[0],
        this.anchor.elements[1] + V[1],
        this.anchor.elements[2] + (V[2] || 0)
      ], this.direction);
    },
  
    // Returns true if the line is parallel to the argument. Here, 'parallel to'
    // means that the argument's direction is either parallel or antiparallel to
    // the line's own direction. A line is parallel to a plane if the two do not
    // have a unique intersection.
    isParallelTo: function(obj) {
      if (obj.normal) { return obj.isParallelTo(this); }
      var theta = this.direction.angleFrom(obj.direction);
      return (Math.abs(theta) <= Sylvester.precision || Math.abs(theta - Math.PI) <= Sylvester.precision);
    },
  
    // Returns the line's perpendicular distance from the argument,
    // which can be a point, a line or a plane
    distanceFrom: function(obj) {
      if (obj.normal) { return obj.distanceFrom(this); }
      if (obj.direction) {
        // obj is a line
        if (this.isParallelTo(obj)) { return this.distanceFrom(obj.anchor); }
        var N = this.direction.cross(obj.direction).toUnitVector().elements;
        var A = this.anchor.elements, B = obj.anchor.elements;
        return Math.abs((A[0] - B[0]) * N[0] + (A[1] - B[1]) * N[1] + (A[2] - B[2]) * N[2]);
      } else {
        // obj is a point
        var P = obj.elements || obj;
        var A = this.anchor.elements, D = this.direction.elements;
        var PA1 = P[0] - A[0], PA2 = P[1] - A[1], PA3 = (P[2] || 0) - A[2];
        var modPA = Math.sqrt(PA1*PA1 + PA2*PA2 + PA3*PA3);
        if (modPA === 0) return 0;
        // Assumes direction vector is normalized
        var cosTheta = (PA1 * D[0] + PA2 * D[1] + PA3 * D[2]) / modPA;
        var sin2 = 1 - cosTheta*cosTheta;
        return Math.abs(modPA * Math.sqrt(sin2 < 0 ? 0 : sin2));
      }
    },
  
    // Returns true iff the argument is a point on the line
    contains: function(point) {
      var dist = this.distanceFrom(point);
      return (dist !== null && dist <= Sylvester.precision);
    },
  
    // Returns true iff the line lies in the given plane
    liesIn: function(plane) {
      return plane.contains(this);
    },
  
    // Returns true iff the line has a unique point of intersection with the argument
    intersects: function(obj) {
      if (obj.normal) { return obj.intersects(this); }
      return (!this.isParallelTo(obj) && this.distanceFrom(obj) <= Sylvester.precision);
    },
  
    // Returns the unique intersection point with the argument, if one exists
    intersectionWith: function(obj) {
      if (obj.normal) { return obj.intersectionWith(this); }
      if (!this.intersects(obj)) { return null; }
      var P = this.anchor.elements, X = this.direction.elements,
          Q = obj.anchor.elements, Y = obj.direction.elements;
      var X1 = X[0], X2 = X[1], X3 = X[2], Y1 = Y[0], Y2 = Y[1], Y3 = Y[2];
      var PsubQ1 = P[0] - Q[0], PsubQ2 = P[1] - Q[1], PsubQ3 = P[2] - Q[2];
      var XdotQsubP = - X1*PsubQ1 - X2*PsubQ2 - X3*PsubQ3;
      var YdotPsubQ = Y1*PsubQ1 + Y2*PsubQ2 + Y3*PsubQ3;
      var XdotX = X1*X1 + X2*X2 + X3*X3;
      var YdotY = Y1*Y1 + Y2*Y2 + Y3*Y3;
      var XdotY = X1*Y1 + X2*Y2 + X3*Y3;
      var k = (XdotQsubP * YdotY / XdotX + XdotY * YdotPsubQ) / (YdotY - XdotY * XdotY);
      return Vector.create([P[0] + k*X1, P[1] + k*X2, P[2] + k*X3]);
    },
  
    // Returns the point on the line that is closest to the given point or line
    pointClosestTo: function(obj) {
      if (obj.direction) {
        // obj is a line
        if (this.intersects(obj)) { return this.intersectionWith(obj); }
        if (this.isParallelTo(obj)) { return null; }
        var D = this.direction.elements, E = obj.direction.elements;
        var D1 = D[0], D2 = D[1], D3 = D[2], E1 = E[0], E2 = E[1], E3 = E[2];
        // Create plane containing obj and the shared normal and intersect this with it
        // Thank you: http://www.cgafaq.info/wiki/Line-line_distance
        var x = (D3 * E1 - D1 * E3), y = (D1 * E2 - D2 * E1), z = (D2 * E3 - D3 * E2);
        var N = Vector.create([x * E3 - y * E2, y * E1 - z * E3, z * E2 - x * E1]);
        var P = Plane.create(obj.anchor, N);
        return P.intersectionWith(this);
      } else {
        // obj is a point
        var P = obj.elements || obj;
        if (this.contains(P)) { return Vector.create(P); }
        var A = this.anchor.elements, D = this.direction.elements;
        var D1 = D[0], D2 = D[1], D3 = D[2], A1 = A[0], A2 = A[1], A3 = A[2];
        var x = D1 * (P[1]-A2) - D2 * (P[0]-A1), y = D2 * ((P[2] || 0) - A3) - D3 * (P[1]-A2),
            z = D3 * (P[0]-A1) - D1 * ((P[2] || 0) - A3);
        var V = Vector.create([D2 * x - D3 * z, D3 * y - D1 * x, D1 * z - D2 * y]);
        var k = this.distanceFrom(P) / V.modulus();
        return Vector.create([
          P[0] + V.elements[0] * k,
          P[1] + V.elements[1] * k,
          (P[2] || 0) + V.elements[2] * k
        ]);
      }
    },
  
    // Returns a copy of the line rotated by t radians about the given line. Works by
    // finding the argument's closest point to this line's anchor point (call this C) and
    // rotating the anchor about C. Also rotates the line's direction about the argument's.
    // Be careful with this - the rotation axis' direction affects the outcome!
    rotate: function(t, line) {
      // If we're working in 2D
      if (typeof(line.direction) == 'undefined') { line = Line.create(line.to3D(), Vector.k); }
      var R = Matrix.Rotation(t, line.direction).elements;
      var C = line.pointClosestTo(this.anchor).elements;
      var A = this.anchor.elements, D = this.direction.elements;
      var C1 = C[0], C2 = C[1], C3 = C[2], A1 = A[0], A2 = A[1], A3 = A[2];
      var x = A1 - C1, y = A2 - C2, z = A3 - C3;
      return Line.create([
        C1 + R[0][0] * x + R[0][1] * y + R[0][2] * z,
        C2 + R[1][0] * x + R[1][1] * y + R[1][2] * z,
        C3 + R[2][0] * x + R[2][1] * y + R[2][2] * z
      ], [
        R[0][0] * D[0] + R[0][1] * D[1] + R[0][2] * D[2],
        R[1][0] * D[0] + R[1][1] * D[1] + R[1][2] * D[2],
        R[2][0] * D[0] + R[2][1] * D[1] + R[2][2] * D[2]
      ]);
    },
  
    // Returns the line's reflection in the given point or line
    reflectionIn: function(obj) {
      if (obj.normal) {
        // obj is a plane
        var A = this.anchor.elements, D = this.direction.elements;
        var A1 = A[0], A2 = A[1], A3 = A[2], D1 = D[0], D2 = D[1], D3 = D[2];
        var newA = this.anchor.reflectionIn(obj).elements;
        // Add the line's direction vector to its anchor, then mirror that in the plane
        var AD1 = A1 + D1, AD2 = A2 + D2, AD3 = A3 + D3;
        var Q = obj.pointClosestTo([AD1, AD2, AD3]).elements;
        var newD = [Q[0] + (Q[0] - AD1) - newA[0], Q[1] + (Q[1] - AD2) - newA[1], Q[2] + (Q[2] - AD3) - newA[2]];
        return Line.create(newA, newD);
      } else if (obj.direction) {
        // obj is a line - reflection obtained by rotating PI radians about obj
        return this.rotate(Math.PI, obj);
      } else {
        // obj is a point - just reflect the line's anchor in it
        var P = obj.elements || obj;
        return Line.create(this.anchor.reflectionIn([P[0], P[1], (P[2] || 0)]), this.direction);
      }
    },
  
    // Set the line's anchor point and direction.
    setVectors: function(anchor, direction) {
      // Need to do this so that line's properties are not
      // references to the arguments passed in
      anchor = Vector.create(anchor);
      direction = Vector.create(direction);
      if (anchor.elements.length == 2) {anchor.elements.push(0); }
      if (direction.elements.length == 2) { direction.elements.push(0); }
      if (anchor.elements.length > 3 || direction.elements.length > 3) { return null; }
      var mod = direction.modulus();
      if (mod === 0) { return null; }
      this.anchor = anchor;
      this.direction = Vector.create([
        direction.elements[0] / mod,
        direction.elements[1] / mod,
        direction.elements[2] / mod
      ]);
      return this;
    }
  };
  
    
  // Constructor function
  Line.create = function(anchor, direction) {
    var L = new Line();
    return L.setVectors(anchor, direction);
  };
  
  // Axes
  Line.X = Line.create(Vector.Zero(3), Vector.i);
  Line.Y = Line.create(Vector.Zero(3), Vector.j);
  Line.Z = Line.create(Vector.Zero(3), Vector.k);
  
  
  
  function Plane() {}
  Plane.prototype = {
  
    // Returns true iff the plane occupies the same space as the argument
    eql: function(plane) {
      return (this.contains(plane.anchor) && this.isParallelTo(plane));
    },
  
    // Returns a copy of the plane
    dup: function() {
      return Plane.create(this.anchor, this.normal);
    },
  
    // Returns the result of translating the plane by the given vector
    translate: function(vector) {
      var V = vector.elements || vector;
      return Plane.create([
        this.anchor.elements[0] + V[0],
        this.anchor.elements[1] + V[1],
        this.anchor.elements[2] + (V[2] || 0)
      ], this.normal);
    },
  
    // Returns true iff the plane is parallel to the argument. Will return true
    // if the planes are equal, or if you give a line and it lies in the plane.
    isParallelTo: function(obj) {
      var theta;
      if (obj.normal) {
        // obj is a plane
        theta = this.normal.angleFrom(obj.normal);
        return (Math.abs(theta) <= Sylvester.precision || Math.abs(Math.PI - theta) <= Sylvester.precision);
      } else if (obj.direction) {
        // obj is a line
        return this.normal.isPerpendicularTo(obj.direction);
      }
      return null;
    },
    
    // Returns true iff the receiver is perpendicular to the argument
    isPerpendicularTo: function(plane) {
      var theta = this.normal.angleFrom(plane.normal);
      return (Math.abs(Math.PI/2 - theta) <= Sylvester.precision);
    },
  
    // Returns the plane's distance from the given object (point, line or plane)
    distanceFrom: function(obj) {
      if (this.intersects(obj) || this.contains(obj)) { return 0; }
      if (obj.anchor) {
        // obj is a plane or line
        var A = this.anchor.elements, B = obj.anchor.elements, N = this.normal.elements;
        return Math.abs((A[0] - B[0]) * N[0] + (A[1] - B[1]) * N[1] + (A[2] - B[2]) * N[2]);
      } else {
        // obj is a point
        var P = obj.elements || obj;
        var A = this.anchor.elements, N = this.normal.elements;
        return Math.abs((A[0] - P[0]) * N[0] + (A[1] - P[1]) * N[1] + (A[2] - (P[2] || 0)) * N[2]);
      }
    },
  
    // Returns true iff the plane contains the given point or line
    contains: function(obj) {
      if (obj.normal) { return null; }
      if (obj.direction) {
        return (this.contains(obj.anchor) && this.contains(obj.anchor.add(obj.direction)));
      } else {
        var P = obj.elements || obj;
        var A = this.anchor.elements, N = this.normal.elements;
        var diff = Math.abs(N[0]*(A[0] - P[0]) + N[1]*(A[1] - P[1]) + N[2]*(A[2] - (P[2] || 0)));
        return (diff <= Sylvester.precision);
      }
    },
  
    // Returns true iff the plane has a unique point/line of intersection with the argument
    intersects: function(obj) {
      if (typeof(obj.direction) == 'undefined' && typeof(obj.normal) == 'undefined') { return null; }
      return !this.isParallelTo(obj);
    },
  
    // Returns the unique intersection with the argument, if one exists. The result
    // will be a vector if a line is supplied, and a line if a plane is supplied.
    intersectionWith: function(obj) {
      if (!this.intersects(obj)) { return null; }
      if (obj.direction) {
        // obj is a line
        var A = obj.anchor.elements, D = obj.direction.elements,
            P = this.anchor.elements, N = this.normal.elements;
        var multiplier = (N[0]*(P[0]-A[0]) + N[1]*(P[1]-A[1]) + N[2]*(P[2]-A[2])) / (N[0]*D[0] + N[1]*D[1] + N[2]*D[2]);
        return Vector.create([A[0] + D[0]*multiplier, A[1] + D[1]*multiplier, A[2] + D[2]*multiplier]);
      } else if (obj.normal) {
        // obj is a plane
        var direction = this.normal.cross(obj.normal).toUnitVector();
        // To find an anchor point, we find one co-ordinate that has a value
        // of zero somewhere on the intersection, and remember which one we picked
        var N = this.normal.elements, A = this.anchor.elements,
            O = obj.normal.elements, B = obj.anchor.elements;
        var solver = Matrix.Zero(2,2), i = 0;
        while (solver.isSingular()) {
          i++;
          solver = Matrix.create([
            [ N[i%3], N[(i+1)%3] ],
            [ O[i%3], O[(i+1)%3]  ]
          ]);
        }
        // Then we solve the simultaneous equations in the remaining dimensions
        var inverse = solver.inverse().elements;
        var x = N[0]*A[0] + N[1]*A[1] + N[2]*A[2];
        var y = O[0]*B[0] + O[1]*B[1] + O[2]*B[2];
        var intersection = [
          inverse[0][0] * x + inverse[0][1] * y,
          inverse[1][0] * x + inverse[1][1] * y
        ];
        var anchor = [];
        for (var j = 1; j <= 3; j++) {
          // This formula picks the right element from intersection by
          // cycling depending on which element we set to zero above
          anchor.push((i == j) ? 0 : intersection[(j + (5 - i)%3)%3]);
        }
        return Line.create(anchor, direction);
      }
    },
  
    // Returns the point in the plane closest to the given point
    pointClosestTo: function(point) {
      var P = point.elements || point;
      var A = this.anchor.elements, N = this.normal.elements;
      var dot = (A[0] - P[0]) * N[0] + (A[1] - P[1]) * N[1] + (A[2] - (P[2] || 0)) * N[2];
      return Vector.create([P[0] + N[0] * dot, P[1] + N[1] * dot, (P[2] || 0) + N[2] * dot]);
    },
  
    // Returns a copy of the plane, rotated by t radians about the given line
    // See notes on Line#rotate.
    rotate: function(t, line) {
      var R = Matrix.Rotation(t, line.direction).elements;
      var C = line.pointClosestTo(this.anchor).elements;
      var A = this.anchor.elements, N = this.normal.elements;
      var C1 = C[0], C2 = C[1], C3 = C[2], A1 = A[0], A2 = A[1], A3 = A[2];
      var x = A1 - C1, y = A2 - C2, z = A3 - C3;
      return Plane.create([
        C1 + R[0][0] * x + R[0][1] * y + R[0][2] * z,
        C2 + R[1][0] * x + R[1][1] * y + R[1][2] * z,
        C3 + R[2][0] * x + R[2][1] * y + R[2][2] * z
      ], [
        R[0][0] * N[0] + R[0][1] * N[1] + R[0][2] * N[2],
        R[1][0] * N[0] + R[1][1] * N[1] + R[1][2] * N[2],
        R[2][0] * N[0] + R[2][1] * N[1] + R[2][2] * N[2]
      ]);
    },
  
    // Returns the reflection of the plane in the given point, line or plane.
    reflectionIn: function(obj) {
      if (obj.normal) {
        // obj is a plane
        var A = this.anchor.elements, N = this.normal.elements;
        var A1 = A[0], A2 = A[1], A3 = A[2], N1 = N[0], N2 = N[1], N3 = N[2];
        var newA = this.anchor.reflectionIn(obj).elements;
        // Add the plane's normal to its anchor, then mirror that in the other plane
        var AN1 = A1 + N1, AN2 = A2 + N2, AN3 = A3 + N3;
        var Q = obj.pointClosestTo([AN1, AN2, AN3]).elements;
        var newN = [Q[0] + (Q[0] - AN1) - newA[0], Q[1] + (Q[1] - AN2) - newA[1], Q[2] + (Q[2] - AN3) - newA[2]];
        return Plane.create(newA, newN);
      } else if (obj.direction) {
        // obj is a line
        return this.rotate(Math.PI, obj);
      } else {
        // obj is a point
        var P = obj.elements || obj;
        return Plane.create(this.anchor.reflectionIn([P[0], P[1], (P[2] || 0)]), this.normal);
      }
    },
  
    // Sets the anchor point and normal to the plane. If three arguments are specified,
    // the normal is calculated by assuming the three points should lie in the same plane.
    // If only two are sepcified, the second is taken to be the normal. Normal vector is
    // normalised before storage.
    setVectors: function(anchor, v1, v2) {
      anchor = Vector.create(anchor);
      anchor = anchor.to3D(); if (anchor === null) { return null; }
      v1 = Vector.create(v1);
      v1 = v1.to3D(); if (v1 === null) { return null; }
      if (typeof(v2) == 'undefined') {
        v2 = null;
      } else {
        v2 = Vector.create(v2);
        v2 = v2.to3D(); if (v2 === null) { return null; }
      }
      var A1 = anchor.elements[0], A2 = anchor.elements[1], A3 = anchor.elements[2];
      var v11 = v1.elements[0], v12 = v1.elements[1], v13 = v1.elements[2];
      var normal, mod;
      if (v2 !== null) {
        var v21 = v2.elements[0], v22 = v2.elements[1], v23 = v2.elements[2];
        normal = Vector.create([
          (v12 - A2) * (v23 - A3) - (v13 - A3) * (v22 - A2),
          (v13 - A3) * (v21 - A1) - (v11 - A1) * (v23 - A3),
          (v11 - A1) * (v22 - A2) - (v12 - A2) * (v21 - A1)
        ]);
        mod = normal.modulus();
        if (mod === 0) { return null; }
        normal = Vector.create([normal.elements[0] / mod, normal.elements[1] / mod, normal.elements[2] / mod]);
      } else {
        mod = Math.sqrt(v11*v11 + v12*v12 + v13*v13);
        if (mod === 0) { return null; }
        normal = Vector.create([v1.elements[0] / mod, v1.elements[1] / mod, v1.elements[2] / mod]);
      }
      this.anchor = anchor;
      this.normal = normal;
      return this;
    }
  };
  
  // Constructor function
  Plane.create = function(anchor, v1, v2) {
    var P = new Plane();
    return P.setVectors(anchor, v1, v2);
  };
  
  // X-Y-Z planes
  Plane.XY = Plane.create(Vector.Zero(3), Vector.k);
  Plane.YZ = Plane.create(Vector.Zero(3), Vector.i);
  Plane.ZX = Plane.create(Vector.Zero(3), Vector.j);
  Plane.YX = Plane.XY; Plane.ZY = Plane.YZ; Plane.XZ = Plane.ZX;
  
  // Utility functions
  var $V = Vector.create;
  var $M = Matrix.create;
  var $L = Line.create;
  var $P = Plane.create;

  // augment Sylvester some
Matrix.Translation = function (v)
{
  if (v.elements.length == 2) {
    var r = Matrix.I(3);
    r.elements[2][0] = v.elements[0];
    r.elements[2][1] = v.elements[1];
    return r;
  }

  if (v.elements.length == 3) {
    var r = Matrix.I(4);
    r.elements[0][3] = v.elements[0];
    r.elements[1][3] = v.elements[1];
    r.elements[2][3] = v.elements[2];
    return r;
  }

  throw "Invalid length for Translation";
}

Matrix.prototype.flatten = function ()
{
    var result = [];
    if (this.elements.length == 0)
        return [];


    for (var j = 0; j < this.elements[0].length; j++)
        for (var i = 0; i < this.elements.length; i++)
            result.push(this.elements[i][j]);
    return result;
}

Matrix.prototype.ensure4x4 = function()
{
    if (this.elements.length == 4 &&
        this.elements[0].length == 4)
        return this;

    if (this.elements.length > 4 ||
        this.elements[0].length > 4)
        return null;

    for (var i = 0; i < this.elements.length; i++) {
        for (var j = this.elements[i].length; j < 4; j++) {
            if (i == j)
                this.elements[i].push(1);
            else
                this.elements[i].push(0);
        }
    }

    for (var i = this.elements.length; i < 4; i++) {
        if (i == 0)
            this.elements.push([1, 0, 0, 0]);
        else if (i == 1)
            this.elements.push([0, 1, 0, 0]);
        else if (i == 2)
            this.elements.push([0, 0, 1, 0]);
        else if (i == 3)
            this.elements.push([0, 0, 0, 1]);
    }

    return this;
};

Matrix.prototype.make3x3 = function()
{
    if (this.elements.length != 4 ||
        this.elements[0].length != 4)
        return null;

    return Matrix.create([[this.elements[0][0], this.elements[0][1], this.elements[0][2]],
                          [this.elements[1][0], this.elements[1][1], this.elements[1][2]],
                          [this.elements[2][0], this.elements[2][1], this.elements[2][2]]]);
};

Vector.prototype.flatten = function ()
{
    return this.elements;
};

function mht(m) {
    var s = "";
    if (m.length == 16) {
        for (var i = 0; i < 4; i++) {
            s += "<span style='font-family: monospace'>[" + m[i*4+0].toFixed(4) + "," + m[i*4+1].toFixed(4) + "," + m[i*4+2].toFixed(4) + "," + m[i*4+3].toFixed(4) + "]</span><br>";
        }
    } else if (m.length == 9) {
        for (var i = 0; i < 3; i++) {
            s += "<span style='font-family: monospace'>[" + m[i*3+0].toFixed(4) + "," + m[i*3+1].toFixed(4) + "," + m[i*3+2].toFixed(4) + "]</font><br>";
        }
    } else {
        return m.toString();
    }
    return s;
}

//
// gluLookAt
//
function makeLookAt(ex, ey, ez,
                    cx, cy, cz,
                    ux, uy, uz)
{
    var eye = $V([ex, ey, ez]);
    var center = $V([cx, cy, cz]);
    var up = $V([ux, uy, uz]);

    var mag;

    var z = eye.subtract(center).toUnitVector();
    var x = up.cross(z).toUnitVector();
    var y = z.cross(x).toUnitVector();

    var m = $M([[x.e(1), x.e(2), x.e(3), 0],
                [y.e(1), y.e(2), y.e(3), 0],
                [z.e(1), z.e(2), z.e(3), 0],
                [0, 0, 0, 1]]);

    var t = $M([[1, 0, 0, -ex],
                [0, 1, 0, -ey],
                [0, 0, 1, -ez],
                [0, 0, 0, 1]]);
    return m.x(t);
}

//
// glOrtho
//
function makeOrtho(left, right,
                   bottom, top,
                   znear, zfar)
{
    var tx = -(right+left)/(right-left);
    var ty = -(top+bottom)/(top-bottom);
    var tz = -(zfar+znear)/(zfar-znear);

    return $M([[2/(right-left), 0, 0, tx],
               [0, 2/(top-bottom), 0, ty],
               [0, 0, -2/(zfar-znear), tz],
               [0, 0, 0, 1]]);
}

//
// gluPerspective
//
function makePerspective(fovy, aspect, znear, zfar)
{
    var ymax = znear * Math.tan(fovy * Math.PI / 360.0);
    var ymin = -ymax;
    var xmin = ymin * aspect;
    var xmax = ymax * aspect;

    return makeFrustum(xmin, xmax, ymin, ymax, znear, zfar);
}

//
// glFrustum
//
function makeFrustum(left, right,
                     bottom, top,
                     znear, zfar)
{
    var X = 2*znear/(right-left);
    var Y = 2*znear/(top-bottom);
    var A = (right+left)/(right-left);
    var B = (top+bottom)/(top-bottom);
    var C = -(zfar+znear)/(zfar-znear);
    var D = -2*zfar*znear/(zfar-znear);

    return $M([[X, 0, A, 0],
               [0, Y, B, 0],
               [0, 0, C, D],
               [0, 0, -1, 0]]);
}

//
// glOrtho
//
function makeOrtho(left, right, bottom, top, znear, zfar)
{
    var tx = - (right + left) / (right - left);
    var ty = - (top + bottom) / (top - bottom);
    var tz = - (zfar + znear) / (zfar - znear);

    return $M([[2 / (right - left), 0, 0, tx],
	       [0, 2 / (top - bottom), 0, ty],
	       [0, 0, -2 / (zfar - znear), tz],
	       [0, 0, 0, 1]]);
}
